2019 MATHEMATICS TEAM CHALLENGESENIOR SECONDARY

TEAMS CONTEST

Time: 45 minutesCalculators may be usedEach question is worth 10 pointsTotal of 100 points

T1 (10 points)Leonhard Euler (1707-1783) was blessed with an amazing memory. He knew the first six powers of the integers from 1 to 100. For example the sixth power of 99 is 941,480,149,401. (996 = 941,480,149,401)

What is the units digit of the sum below?

16 + 26 + 36 + 46 + 56 +…….. + 9976 + 9986 + 9996

T2 (10 points)What is the largest square number which is also a factor of 10!? [Note 10! = 10 × 9× 8 ×……× 3× 2× 1]

T3 (10 points) The World Series of Baseball is a best 4 out of 7 series. This means both teams play until one team has won four games. Therefore a series could last 4, 5, 6 or 7 games. What is the probability that the World Series is at least 6 games?

T4 (10 points) Leonardo the blind painter used four different coloured tins of paint to paint his masterpiece. After he had finished, he randomly placed the lids back on his four tins of paint. What are the chances (as a percent) that none of the lids was placed on its correct tin?

T5 (10 points)A number is abundant if the sum of its factors (not including the number itself) is greater than the number.For example, 18 is abundant, since its factors 1, 2, 3, 6, 9 add to 21. What is the probability (as a percentage) that a number selected at random out of the first 100 positive integers is abundant?

T6 (10 points)What is the length in metres of d in the quadrilateral below?

T7 (10 points)The magic square below uses base 2 or binary. For example, 5 in binary is 101. What is the binary representation of the number N in the magic square ?

T8 (10 points)Find the only positive integer value for a such that a2+2 is divisible by 2a+3.

T9 (10 points)The Tower of Hanoi puzzle involves moving discs of distinct sizes between three rods. The puzzle starts with discs neatly stacked in order of size on one rod, with the smallest on top as shown in the diagram. The goal is transfer the entire stack to another rod with the smallest on top with the least number of moves. The rules are that:

no disc can be placed on top of a smaller disc during the transfer, and only one disk can be moved each time

Fifteen moves are required to move the four discs shown below to another rod. What is the smallest number of moves required to move a stack of 10 discs to another rod?

T10 (10 points)The Akashi Kaikyo Bridge in Japan is the longest suspension bridge in the world. The cable suspended between the two main towers is a catenary curve that can be modelled by a parabola with equation y = 0.000188x2 + 15, where y is the height (in metres) above the road and x is distance (in metres) from the centre of the bridge.

If the tops of the two towers are 201 metres above the bridge, what is the distance (rounded to the nearest whole metre) between the main towers ?

2019 MATHS TEAM CHALLENGE – YEARS 11-12 TEAM EVENT

RESPONSE SHEET

Q Response Points Score

T1 10

T2 10

T3 10

T4 10

T5 10

T6 10

T7 10

T8 10

T9 10

T10 10

School Team Total Score

2019 MATHS TEAM CHALLENGESENIOR SECONDARY

TEAM EVENT

ANSWER SHEET

Question Answers Points

T1. (10 points) 0

T2. (10 points) 518,400

T3. (10 points) 58 OR 62.5% OR 0.0625

T4. (10 points) 37.5%

T5. (10 points) 22%

T6. (10 points) 4 metres

T7. (10 points) 1001

T8. (10 points) a = 7

T9. (10 points) 1023

T10. (10 points) 1,989 metres